If $n$ is an odd integer and $b$ is an integer prime with $n$, state what it means for $n$ to be a pseudoprime to the base $b$. What is a Carmichael number? State a criterion for $n$ to be a Carmichael number and use the criterion to show that:

(i) Every Carmichael number is the product of at least three distinct primes.

(ii) 561 is a Carmichael number.